Higher-order shear and normal deformable theory for functionally graded incompressible linear elastic plates

doi:10.1016/j.tws.2007.07.008

Thin-Walled Structures

Volume 45, Issue 12, December 2007, Pages 974-982

Dedicated to my friend and colleague Professor M.W. Hyer on his 65th birthday

https://doi.org/10.1016/j.tws.2007.07.008 Get rights and content

Abstract

We use the principle of virtual work to derive a higher-order shear and normal deformable theory for a plate comprised of a linear elastic incompressible anisotropic material. The theory does not use a shear correction factor and employs three components of displacement and the hydrostatic pressure as independent variables. For a $K$ th order plate theory, a set of $4 (K + 1)$ coupled equations need to be solved for the $(K + 1)$ pressures and the $3 (K + 1)$ displacements defined on the reference surface of the plate.

Equations for free vibrations of a plate are derived, and equations for the determination of frequencies and the corresponding mode shapes of a simply supported rectangular plate are given.

Introduction

Because of the increasing interest in rubberlike materials and elastomers and their use in automotive and aerospace industries we develop here a higher-order shear and normal deformable theory for plates made of incompressible linear elastic materials. Only isochoric or volume preserving deformations are admissible in an incompressible material. Accordingly, the constitutive relation for an incompressible material involves a hydrostatic pressure that cannot be determined from the strain field but is found by solving equations governing deformations of the body and the prescribed initial and boundary conditions. The pressure field can be determined uniquely only if normal surface tractions are prescribed on a part of the boundary of the body. Thus for a plate made of an incompressible material we need not only equations to find the displacement field but also the pressure field.

Aimmanee and Batra [1] have recently given an analytical solution for free vibrations of a simply supported rectangular plate made of an incompressible linear elastic material. For a thick plate their results show that the displacement and the pressure fields do not vary linearly through the plate thickness. Also, for a thick plate frequencies of some in-plane modes of vibration with the lateral deflection identically zero are lower than those of the out-of-plane modes of vibration with non-vanishing lateral deflections. We develop here a plate theory in which the three displacement components and the pressure are expanded in Taylor series in the thickness coordinate, z, and terms of the same degree in z are retained in their expansions. Since both transverse shear and transverse normal strains are considered, we call the theory shear and normal deformable. The plate theory is called $K$ th order if terms of order $z^{K}$ are kept in the Taylor series expansions in z of displacements and the pressure field. The order of the theory suitable for a plate depends upon the ratio, R, of the plate thickness to the characteristic in-plane dimension and which aspects of the 3-dimensional (3-D) deformations are to be well approximated; a small value of K suffices for a thin plate with $R ⪡ 1$ .

Batra and Vidoli [2] followed Mindlin and Medick [3] in using Legendre polynomials in z as basis functions to derive a mixed higher-order shear and normal deformable plate theory (HOSNDPT) for piezoelectric plates from a mixed variational principle [4]. Subsequently, Batra et al. [5] employed the Reissner–Hellinger mixed principle to deduce a mixed HOSNDPT and also a compatible HOSNDPT. Whereas in the former independent expansions for displacements and stresses are presumed, in the latter only displacements are expanded. Strains deduced from the displacement field and the pertinent constitutive relation are used to find stresses. Here we employ the principle of virtual work to derive a compatible HOSNDPT for a plate made of an incompressible anisotropic linear elastic inhomogeneous or functionally graded material. We do not require that the transverse normal and/or the transverse shear strains must vanish on the top and the bottom surfaces of a plate. Thus the effect of tangential tractions applied on these surfaces can be studied.

There are numerous papers on plate theories. However, they assume the plate material to be compressible. Herrmann [6] reorganized equations of linear elasticity so that they are applicable to both compressible and incompressible materials; these also involve the three components of displacement and the pressure as unknowns. Other works that consider transverse normal and transverse shear strains include those of Mindlin and Medick [3], Vlasov [7], Lo et al. [8], Kant [9], Hanna and Leissa [10], Reddy [11], Lee and Yu [12], and Lee et al. [13]. This list is by no means complete since the number of papers on plate theories is too large to be included here.

Section snippets

Formulation of the problem

We use rectangular Cartesian coordinates to describe infinitesimal deformations of a plate made of an incompressible linear elastic material. Let the $x_{1} x_{2}$ -plane coincide with the mid-surface of the plate and the $x_{3}$ -axis be along the thickness direction. With h equaling the plate thickness, we normalize lengths by $h / 2$ so that $x_{3} =+ 1$ and $- 1$ at points on the top and the bottom surfaces, $S^{+}$ and $S^{-}$ , respectively, of the plate.

Equations governing 3-D infinitesimal deformations of a linear elastic

Derivation of the higher-order plate theory

Let $L_{0} (z), L_{1} (z), L_{2} (z), \dots$ be orthonormal Legendre polynomials satisfying $\int_{- 1}^{1} L_{a} (z) L_{b} (z) d z = δ_{ab}, a, b = 0, 1, 2, \dots, K .$ Note that $L_{0} (z) = \frac{1}{\sqrt{2}}, L_{1} (z) = \sqrt{\frac{3}{2}} z, L_{2} (z) = \sqrt{\frac{5}{2}} (\frac{3 z^{2}}{2} - \frac{1}{2}), L_{3} (z) = \sqrt{\frac{7}{2}} (- 3 \frac{z}{2} + \frac{5}{2} z^{3}), \dots .$ The basis functions $L_{0} (z), L_{1} (z), \dots, L_{K} (z)$ are equivalent to $1, z, z^{2}, \dots, z^{K}$ , and are, alternatively, even and odd functions of z. An advantage of using orthonormal basis functions is that the algebraic work is reduced. Henceforth we set $x_{3} = z$ , and $u_{i} (x_{1}, x_{2}, z, t) = v_{α} (x_{1}, x_{2}, z, t) δ_{i α} + w (x_{1}, x_{2}, z, t) δ_{i 3}, α = 1, 2 .$ That is, $v_{α}$ denotes the in-plane

Balance laws

Let $η_{i}$ be a smooth virtual displacement field defined on $Ω$ that vanishes on $\partial_{u} Ω$ , and $λ$ be a smooth scalar field defined on $Ω$ . Taking the inner product of both sides of Eq. (1) with $η$ , multiplying both sides of Eq. (2) with a Lagrange multiplier function $λ$ , and integrating the resulting equations over $Ω$ , we obtain $\int_{S} d A \int_{- 1}^{1} η_{i} σ_{ij, j} d z + \int_{S} d A \int_{- 1}^{1} ρ b_{i} η_{i} d z = \int_{S} d A \int_{- 1}^{1} ρ {\ddot{u}}_{i} η_{i} d z,$ $\int_{S} d A \int_{- 1}^{1} λ u_{i, i} d z = 0,$ where the area integration is over the midsurface S of the plate. We assume that $η_{i} (x_{1}, x_{2}, z) = L_{a} (z) η_{i}^{a} (x_{1}, x_{2}),$ $λ (x_{1}, x_{2},$

Free vibrations

When studying free vibrations of a plate, we take ${\bar{t}}_{i} = 0$ on $\partial_{t} Ω$ , and $b = 0$ . Thus Eqs. (27), (28) and (25) become ${\hat{M}}_{α β, β}^{a} - p_{, α}^{a} - D_{ab} T_{α}^{b} = R_{ab} {\ddot{v}}_{α}^{b},$ $T_{α, α}^{a} + D_{ab} p^{b} - D_{ab} {\hat{T}}_{3}^{b} = R_{ab} {\ddot{w}}^{b},$ $v_{α, α}^{a} + D_{ca} w^{c} = 0 .$ Substitution for ${\hat{M}}_{α β}^{a}$ , ${\hat{T}}_{3}^{a}$ and $T_{α}^{a}$ from Eq. (30) into Eq. (54) gives $(C_{α β kl} e_{kl}^{a})_{, β} - p_{, α}^{a} - D_{ab} C_{α 3 kl} e_{kl}^{b} = R_{ab} {\ddot{v}}_{α}^{b},$ $(C_{α 3 kl} e_{kl}^{a})_{, α} + D_{ab} p^{b} - D_{ab} C_{33 kl} e_{kl}^{b} = R_{ab} {\ddot{w}}^{b},$ $v_{α, α}^{a} + D_{ca} w^{c} = 0 .$ Henceforth we assume that the plate is made of a homogeneous and isotropic material. Furthermore, we seek solutions of Eq. (55) of the following form: $v_{α}^{a} (x$

Remarks

It is very likely that plates made of rubberlike materials will experience large strains. Thus a plate theory for these materials should incorporate both geometric and material nonlinearities. A possibility is to divide the load into several increments, and for each incremental load develop a theory of infinitesimal deformations of pre-stressed shells since an initially flat plate will be deformed into a part of a shell by the first incremental load. Alternatively, one can use a total

Conclusions

We have used the principle of virtual work to derive a compatible higher-order shear and normal deformable theory for a plate made of an incompressible linear elastic material. The difference between compressible and incompressible materials is that only volume preserving deformations are admissible in incompressible materials, and one needs to find the hydrostatic pressure as a part of the solution of the pertinent initial boundary value problem. This is reflected in the plate theory by also

Acknowledgment

The work was partially supported by the ONR Grant N00014-06-1-0567 to Virginia Polytechnic Institute and State University with Dr. Y.D.S. Rajapakse as the program manager. Views expressed herein are those of the author and neither of the funding agency nor of VPI&SU.

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Higher-order shear and normal deformable theory for functionally graded incompressible linear elastic plates

Abstract

Introduction

Section snippets

Formulation of the problem

Derivation of the higher-order plate theory

Balance laws

Free vibrations

Remarks

Conclusions

Acknowledgment

J Sound Vib

Comput Methods Appl Mech Eng

J Sound Vib

J Sound Vib

Int J of Solids Struct

J Sound Vib

J Sound Vib

J Sound Vib

Comput Struc

Higher order piezoelectric plate theory derived from a three-dimensional variational principle

AIAA J

Extensional vibrations of thick plates

J Appl Mech